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Microtubules are structures within the cell that form a transportation network along which motor proteins tow cargo to destinations. To establish and maintain a structure capable of serving the cell’s tasks, microtubules undergo deconstruction and reconstruction regularly. This change in structure is critical to tasks like wound repair and cell motility. Images of fluorescing microtubule networks are captured in grayscale at different wavelengths, displaying different tagged proteins. The analysis of these polymeric structures involves identifying the presence of the protein and the direction of the structure in which it resides. This study considers the problem of finding statistical properties of sections of microtubules. We consider the research done on directional filters and utilize a basic solution to find the center of a ridge. The method processes the captured image by centering a circle around pre-determined pixel locations so that the highest possible average pixel intensity is found within the circle, thus marking the center of the microtubule. The location of these centers allows us to estimate angular direction and curvature of the microtubules, statistically estimate the direction of microtubules in a region of the cell, and compare properties of different types of microtubule networks in the same region. To verify accuracy, we study the results of the method on a test image.

Microtubules are part of the cellular cytoskeleton and one of their main functions is to serve as a transportation network within the cell. They are made of two subunits: a- and b-tubulin. A microtubule is typically formed by thirteen protofilament chains made of these a-b heterodimers, arranged into a hollow tube approximately 25 mm in diameter [

In the cell, it is typically not necessary for individual microtubules to have a precise subcellular localization. Rather, microtubule networks function normally if they are sufficiently dense and enough segments of micro- tubules have the appropriate orientation. By segments, we mean parts of a mictrotubule along the entire length of the microtubule. These segments could be defined in several ways. In this study, we consider segments of microtubules between where microtubules overlap or cross.

A current problem in microtubule research is determining appropriate statistical measures that quantify the distribution and organization of microtubule networks. Appropriate statistical measures that describe the overall organization of the microtubule network and how the network changes under different conditions can lead to a better understanding of how the cell’s transportation network functions and how it responds to varying external conditions. Two measures that will help in understanding the microtubule network are angle direction and curvature. A distribution of angle direction of segments of microtubules will show the general direction of the microtubules. For instance, a travelling cell will have more microtubule segments oriented in the direction of travel. A distribution of curvature of the segments of microtubules will indicate changes in the orientation of segments of microtubules.

There is a large body of work studying the detection of the direction of ridges, edges, and lines; see [

In this study, we propose a simple method to approximate the center of a microtubule by calculating average pixel intensity within a circle translated across the microtubule. This method has the benefit of being simple to understand conceptually. In the first part of this paper a method is developed to determine the center of a digital representation of fluorescing microtubules. With this center, an approximation for angle direction is possible. In the second part of this paper, we calculate curvature by finding three neighboring centers on a microtubule. Using both of these measures obtained from the method presented, we can better understand the organization of the microtubule network.

The goal of developing this method is to distinguish between microtubule network organizations under different cellular conditions, or to distinguish between different subpopulations of microtubules, such as those in which the tubulin subunits are post-translationally modified, by using a consistent and accurate technique for analyzing images of the microtubule network. The performance of the method will be studied using constructed “test” images with representations of microtubule-like structures in the form of circles, ellipses, and lines. The data collected from applying the method to these constructed microtubules can be compared to theoretical values to analyze accuracy.

There are different approaches to microtubule network structure. One is to map individual microtubules and collect appropriate statistics on direction. A second is to determine a statistical description of microtubule angles along with a probability density of microtubule presence. Here, an approach is adopted that obviates the need to map whole microtubules, but adds information on direction changes to the statistical approach. This is done by considering segments of microtubules and the distributions of angle direction and curvature of these segments. The probability of a microtubule with a determined direction is calculated within an analyzed area of the cell. This approach may produce a more rigorous, statistical analysis of microtubule network structure to determine any differences between different networks of microtubules.

Many situations involve spatial structures whose exact locations are of less concern than their density. In addition, structure directions are often of interest. Microtubules appear on the cellular scale as one-dimensional structures spanning large portions of a cell. Microtubule position and direction are determined by molecular scale processes such as dynamic instability, which could allow the structures to change in response to different conditions.

For the purposes of this study, we assume that a probabalistic description is appropriate. Thus, we seek a probability density function

The theoretical procedure for estimating probabilities is to sample many cells from the set of equivalent cells, locate the point

There is much image data that is essentially two dimensional. In this case, the probability identity is

Additionally, it is difficult to obtain samples of “equivalent” cells when the cells are not symmetric. In this case, we shall consider sub-areas of images that are nearly equivalent. For example, in

where

where

What remains to be found is the conditional probability of microtubule direction

Images of microtubules are two-dimensional grayscale images taken at several vertical positions through the cell (optical sections). Although the cell is three-dimensional, these cells are relatively flat (<5 mm tall). However, we typically take 8 - 15 optical sections per cell to maximize resolution, and then combine these layers using a maximum projection algorithm with image manipulation software such as Image J. Microtubules are labeled with a fluorescently tagged antibody to the tubulin subunit. The distribution of these antibodies along micro- tubules is generally considered to be uniform. These fluorescent molecules create the brighter pixel intensities shown in

Before finding a point near the center of the microtubule we first provide a loose definition of the center of a microtubule. Microtubules are cylindrical and a two-dimensional representation of them would allow for the center of the microtubule to be equi-distant from each side of the two-dimensional representation. However, the images we have of microtubules are, as mentioned elsewhere, actually images of fluorescents attached to tubulin subunits of the microtubule in an assumed uniform distribution. Because of this, the sides of the profile of a microtubule are difficult to define and thus so is the center in this way. Instead we define the center of a microtubule as it affects the curvature of that microtubule segment. A point is on the center of a microtubule if the curvature of the microtubule segment at that point is equal to the real curvature of the microtubule segment. For example, if the microtubule segment were straight and thus had a curvature of 0, a point on the microtubule segment is on the center if the angle between each direction (forward and backward) of the microtubule segment is p and thus the curvature would be 0.

To find points on the microtubule that are close to the center, we find local maxima in the rows and columns of pixel data. These local maxima are at pixel locations with pixel intensity above that of neighboring pixels (in the row only or column only) by a specified amount d. Points found this way are then filtered so that no two points are within one pixel of each other; this avoids double counting at that pixel location. Using this process with a reasonable choice for d finds initial points on nearly all microtubules quickly. The process avoids picking initial points by hand which can create bias, or picking points randomly which often will not be well-centered on the microtuble.

image of ellipse shapes simulating microtubules (described in detail below).

From these points, we find points “centered” on the microtubule. We accomplish this by first finding the angle direction of the microtubule at that location and then translating a circle along the perpendicular angle direction. At many steps along this line, we calculate the average pixel intensity for pixels within the circle. We consider the “center”of the microtuble to be at the center of the circle with the highest possible average pixel intensity. We describe the technique used to find angle direction and the “centering” technique in more detail below.

To implement the centering technique, we first consider that pixel values can be mapped to integer locations on a Cartesian plane, where location is determined by the matrix location of the pixel in the image. For example, pixel

1) Pick, as an initial point, a point with local maximum pixel intensity as discussed earlier.

2) From this initial point, calculate the angle direction of the microtubule.

3) Create a circle-shaped closed constraint centered on this point and translated perpendicular to the found angle direction through n steps in both directions. At each translation step, calculate the average pixel intensity within the circle-shaped closed constraint.

4) The center of the circle corresponding to the step with highest average pixel intensity is considered the center of the microtubule.

5) Repeat for a new initial point.

This process begins with an approximation to the microtuble center by first considering the local row and column maxima and then “centering” a circle on the microtubule, using angle direction and the circle as a guide. This process could be repeated multiple times by recalculating the angle direction at the new “center” and then re-centering the circle. For our proposes, we did not repeat the process.

From these centered points along the microtubule we are able to estimate curvature on test images. Im- plementing this technqiue requires two components not yet discussed: a way to calculate angle direction at a location, and a way to find neighboring triples of points on the microtubule for calculating curvature. To calculate angle direction, the steerable filters discussed above are a well-researched option. We explored another way to calculate angle direction and find neighboring triples of points using a technique similar to the centering

technique described above.

To measure the direction angle of a digital representation of a curve, we must measure angle based on the location of pixels that represent that curve. However measuring direction angle as the slope between two pixels representing points on a curve will result in discrete angles. We can achieve a higher accuracy by measuring the line, or curve, generated from multiple pixels. For images of microtubules, where the pixelated curve has a higher intensity in the center of the curve and lower intensity further from the center, we can consider a two- dimensional area of pixels to determine the direction angle of the curve at a point. We implemented a similar technique to the centering technique described above, using a rotated ellipse-shaped constraint instead of a translated circle. The angle determination method is as follows:

1) Pick an initial point on a microtubule.

2) Create a closed constraint shape centered on this point, and rotate the shape through angle p using t rotation steps. At each rotation step, calculate the average pixel intensity within the closed constraint shape.

3) The angle of rotation of the constraint shape corresponding to the highest average pixel intensity is con- sidered the angle direction of the microtubule at the initial point.

4) Repeat for a new initial point.

The rotated shape with the highest average pixel intensity will best fit over a given section of the fluorescing microtubule, distinguished by bright pixels in the image. See

The problem at each initial point can be written as an integer program:

where the

where the parameters m and n are chosen by trial and error so that the closed shape fits well over the microtubule without being too small or too large that data is inaccurate. The term

If the closed constraint C is offset from the initial point, instead of centered on it, then the resulting maximum average pixel intensity would indicate the approximate direction of travel of the microtubule (see

We apply the method as described to several test images simulating microtubules following curves with known functions and curvatures: ellipses, circles, and lines. These test images are shown in

The constraints detailed above only consider intensities at integer points satisfying the constraint. Those points correspond to the location of a pixel in the image. However, the light entering the aperture of the microscope to be recorded as the pixel intensity can be associated with a two-dimensional area in the image, centered around

the integer point. Because of this, the pixel value of an integer point can be used to represent the pixel value in a unit square area around the integer point. In this situation, the intensity of the entire square area is considered to be the pixel value at the integer point and the intensity of a fraction of the square area is assumed to be the same fraction of the intensity. For example, the intensity of half the square area is considered to be half the pixel value at the integer point. With a method that only considers integer points instead of area, the pixels near or on the border of the contraint will be entirely included or entirely excluded, when only a fraction of their pixel intensity is inside of the constraint. This issue creates a convex hull of the integer points that differs from the real shape of the constraint, and so could lead to inaccuracies in the direction chosen by the method. See Figures 9-16.

To represent the constraint with more accuracy, fractions of pixel values must be approximated near the border of the constraint. To accomplish this, the image can be resized so that each pixel intensity is represented by an

not all, of the resulting resized pixels in the new image. This represents a fraction of the original pixel intensity and allows a better approximation of the constraint. Resizing the image results in a longer computation time, so a trade-off between computation and accuracy is necessary.

Several parameters in the method affect the accuracy, consistency, or useability of the method. These parameters include the resize value, the size and shape of the closed constraint shape, the size of the increments by which the constraint is rotated, and the amount by the which the constraint is offset from the center of rotation. The size of the closed constraint shape (determined by parameters n and m in the constraint equation) were varied to study their effects on the method. To test for changes in the size of the ellipse constraint shape, the offset value, d, was set to one-half the length of the major axis, while the lengths of the minor and major axes were varied from (4, 4) to (10, 20). A resize value of 64 was used with 36,000 rotation steps between 0 and 2p. We expect that a constraint shape too small will not accurately distinguish between similar optimal angle directions, while a constraint shape too large will “jump” from one microtubule to another. Larger constraint shapes require more computing time so a smaller shape is preferential. The largest of the five circle paths (with radius 50 pixels) is used to find curvature at the thirty-six starting points. The results are shown in

curvature of 0.02. The figure on the bottom shows the variance between the starting points scaled so the dif- ferences can be seen better. In each graph, the axes represent the major and minor axes lengths of the ellipse constraint shape, where the major axis is always greater than or equal to the minor axis. Note that error decreases significantly when the ellipse is widened.

To test the improvements to accuracy of the method due to increases in the resize value, a test was designed to exploit starting points at fractions of a pixel. A line-shaped microtubule was used so that the method would have no variation between starting points other than pixel locations. The line used was the first line from the left in

An important check for the method is invariance under rotation and translation of the image. To test for invariance under translation, the test image of circles was reproduced by shifting the generating equation by one- half of a pixel value, which will create a slightly different circle. To test for invariance under rotation, the test image of ellipses was reproduced by adding rotation to the generating equation and creating images rotated from

0 to p in increments of

rotation of the image if the resulting data of these new images is similar to the corresponding data of the original two test images. Points are found and curvature is measured as described in the process above. A Kolmogorov- Smirnov test was used to check differences between two distributions. Examples of the resulting distributions can be seen in

With five circles, five translated circles, and five ellipses rotated eighteen ways, there are 4950 non-trivial comparisons of two distributions. Of these, 287 (5.80%) of the comparisons resulted in a false negative or false positive, given

Four of the five translated circles are indistinguishable from their counterparts, so we consider the method to be sufficiently invariant under translation.

Given that the curvature of a circle is constant and the curvature of an ellipse varies, we may consider only the means and variances of the curvature distributions as a way of separating different shapes.

Thus far we demonstrated that the proposed method for detecting angle direction and curvature at locations of digital curves works with reasonable accuracy and variability of the data. We also showed that the method produces similar results despite differing orientations of the same curve. For these results, angle direction and

curvature are always considered separately. However, angle direction and curvature at a location may not be independent. For instance, in the ellipses shown in

also have angle direction closer to 0 than to

the same figure, we expect curvature to be independent of angle direction. Note that angle direction in these distributions was adjusted by subtracting the mean of the original data. By adjusting in this way, the dis- tributions for rotated ellipses are similar, and so we see that the method gives independence of curvature and angle direction as we expect.

Indepence of curvature and angle direction is important, as then the probability density function,

With an understanding of the performance of the method, we can apply the method to the section of microtubules shown in

angle over

shown in

The distribution of curvature is what we use as the conditional probability density for curvature, given microtubule presence. Applying the probability of microtubule presence as described to this distribution creates the probability of microtubule curvature at a location. The probability density of microtubule presence is shown in

This method maps the high pixel intensity values of an image and reliably records microtubule curvature except at crossings. The resulting data may be used to show a distinction between microtubule subsystems, just as it

shows in this study a distinction between several simple paths. This method can be used to determine the orientation of segments of the microtubule network and then to analyze changes in the microtubule network during events such as cell motility or wound repair. This method can also map, find angle direction, and find curvature of other networks such as blood vessels in the eye, tree limbs, hiking trails, roads, and river deltas. Parameters will need to be adjusted to suit the image and expected properties of the network to ensure accurate results.

Throughout the above research, one issue is mentioned repeatedly that would produce inaccurate results for our purposes if not resolved. This is the issue of finding the center of ridges, or microtubules in our case. In [

The parameters of this method were chosen during testing to ensure more accurate results, but more analysis of the parameters is needed to find a best combination. Most notably, the resize value, fineness of the rotation step, the parameters determining the width and length of the constraint ellipse and the offset value play a crucial role in gathering useful data. The width and length of the constraint ellipse were varied to study the data gathered on the circles of the test image, but the amount by which the ellipse was offset from the current point was not varied. Empirically, the major axis of the ellipse played a major role in generating poor results when too large, while the minor axis of the ellipse helped generate better results when larger.

With this method as a starting point, we hope to establish a process for calculating the curvature of a digital curve at a specified loction along the curve. Several issues must be resolved to create a more useful and reliable method. One of these is computation time associated with a high resize and a fine mesh for the rotation steps. Another issue is determining the change in arc length that the closed constraint shape uses to find the corres- ponding change in angle. Lastly, we must study the differences between the true curve and the digital represen- tation of the curve, in relation to choosing initial points for the method. This method can be used for three- dimensional situations, such as structures of cells, after changes to the structure of the closed constraint shape and the search area are made.

This work was funded by National Institutes of Health grant 5R01GM098619. The cells were processed and photographed by Dr. Lee Ligon.

Tyson DiLorenzo,Lee Ligon,Donald Drew, (2016) Determination of Statistical Properties of Microtubule Populations. Applied Mathematics,07,1456-1475. doi: 10.4236/am.2016.713125